> the image of zero must be zero, and that doesn't hold for (most) affine functions

It does hold, always, in the higher dimension. And I feel like that's the most important thing to clearly understand about affine transforms. The entire beauty of the augmented matrix is that you get a class of non-linear transforms in 3d by using linear transforms in 4d. The article was nice, I'm nit picking something that was nearly there. It'd just be nice to be one teensy bit more explicit about what's going on here.